<< Graphics`Graphics`

x[t_] := a (1 + Cos[t]);
y[t_] := a Sin[t];
Px[θ_] := 2  a - x[θ];
Py[θ_] := 
        y[θ] (((2  a)/x)[θ] - 
              1);

a = 5;
θ = (π/4);

CIRCLE := 
        ParametricPlot[{x[t], y[t]}, {t, 0, 2  π}, 
          PlotRange -> {{-1, 2  a + 1}, {-a - 1, a + 1}}, 
          AspectRatio -> 1];
P := {Px[θ], Py[θ]};
A := {2  a - Px[θ], y[θ]};
B := {2  a, Py[θ] + y[θ]};
ORIGIN := {0, 0};

DisplayTogether[{CIRCLE, 
          Graphics[{Line[{{2  a, -a - 1}, {2  a, 
                    a + 1}}], 

Disk[P, 
                0.1], Disk[A, 0.1], Disk[B, 0.1], 
              Disk[ORIGIN, 0.1]}]}];